Felt is a woolen fabric made by compacting fibers together: it is non-woven. We all know what it looks like, and because this look can be imparted to woven cloth by teaseling* it, which we call felting, there has arisen a tendency to think of felt as woven. Properly speaking though it is not. The website How products are made has a full description.
I recently received a query as to what printer’s felt might be. It was a blanket, a sort of padding used to soften the impression in letterpress printing which facilitated the transfer of the ink to the paper by making the contact less smash-bang rigid. The material used might at sometime have really been felt, but was more commonly something else: paper, cardboard, anything with a bit of give in it.
We come across the word more commonly in paper-making, where a piece of felt is used as a divider between sheets of paper as they were hand delivered by the vatman. The coucher (pronounced coocher) is the one who deploys the felt, as described in Paper making by hand 2.
The felt side is that side of a sheet of paper that has not been in contact with the Fourdrinier wire, and which therefore is the smoother side of the sheet. In modern commercially-made paper this distinction is hard to see, except in the case of a laid paper.
* A teasel is a thistle-like plant of the family Dipsacus. It has hooked prickles and when the flowers are dead the plants are harvested and used in the cloth trade to raise the nap on cloth. When I was a boy woolen mills had huge frames on which hundreds of teasles were mounted. No doubt we nowadays have some man-made cheaper equivalent.